Degeneralization Algorithm for Generation of Büchi Automata Based on Contented Situation

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/jam/volume-2015/516104/figures/516104.fig.002a.svgz" name="516104.fig.002a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>TGBA corresponding to <svg height="12.6787pt" id="M273" style="vertical-align:-3.571951pt" version="1.1" viewbox="-0.0498162 -9.10675 65.7872 12.6787" width="65.7872pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M703 311H403V276C508 267 513 259 513 189V94C513 63 505 47 490 38C477 29 454 26 434 26C270 26 196 184 196 335C196 515 300 623 420 623C514 623 589 576 613 464L651 472C643 550 638 603 635 637C599 645 522 665 440 665C229 665 44 546 44 318C44 117 193 -15 425 -15C506 -15 593 7 653 21C648 47 647 79 647 115V195C647 262 653 266 703 276V311Z" id="g13-69"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,9.85,0)"><path d="M524 497C519 556 514 633 513 650H40V615C113 608 122 601 122 511V142C122 48 113 42 40 35V0H351V35C264 42 256 48 256 142V300H338C427 300 437 292 448 224H484V423H448C437 359 427 350 338 350H256V571C256 604 260 606 295 606H345C401 606 428 599 449 577C466 559 479 529 489 491L524 497Z" id="g13-68"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="554" vert-adv-y="554"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,17.355,0)"><path d="M559 271C559 395 493 448 433 448C370 448 304 394 275 297C259 245 235 121 219 37C157 54 113 110 113 201C113 310 180 386 276 429L265 457C131 410 23 326 23 186C23 83 86 5 211 -10C180 -159 167 -220 158 -252L165 -261C198 -255 236 -246 251 -218L287 -5C421 9 559 119 559 271ZM484 273C484 144 400 40 293 33L352 331C364 389 382 406 405 406C431 406 484 370 484 273Z" id="g113-253"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="582" vert-adv-y="582"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,28.27,0)"><path d="M574 0L325 548H263L14 0H76L293 479H295L512 0H574Z" id="g117-63"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,39.25,0)"><path d="M703 311H403V276C508 267 513 259 513 189V94C513 63 505 47 490 38C477 29 454 26 434 26C270 26 196 184 196 335C196 515 300 623 420 623C514 623 589 576 613 464L651 472C643 550 638 603 635 637C599 645 522 665 440 665C229 665 44 546 44 318C44 117 193 -15 425 -15C506 -15 593 7 653 21C648 47 647 79 647 115V195C647 262 653 266 703 276V311Z" id="g13-69"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,49.1,0)"><path d="M524 497C519 556 514 633 513 650H40V615C113 608 122 601 122 511V142C122 48 113 42 40 35V0H351V35C264 42 256 48 256 142V300H338C427 300 437 292 448 224H484V423H448C437 359 427 350 338 350H256V571C256 604 260 606 295 606H345C401 606 428 599 449 577C466 559 479 529 489 491L524 497Z" id="g13-68"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="554" vert-adv-y="554"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,56.606,0)"><path d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z" id="g113-245"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="654" vert-adv-y="654"></glyph.data></g></svg></td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/jam/volume-2015/516104/figures/516104.fig.002b.svgz" name="516104.fig.002b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Degeneralization of the TGBA</td></tr><tr class="fig-image" id="c"><td><object data="https://static.hindawi.com/articles/jam/volume-2015/516104/figures/516104.fig.002c.svgz" name="516104.fig.002c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>BA corresponding to <svg height="12.6787pt" id="M274" style="vertical-align:-3.571951pt" version="1.1" viewbox="-0.0498162 -9.10675 65.7872 12.6787" width="65.7872pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M703 311H403V276C508 267 513 259 513 189V94C513 63 505 47 490 38C477 29 454 26 434 26C270 26 196 184 196 335C196 515 300 623 420 623C514 623 589 576 613 464L651 472C643 550 638 603 635 637C599 645 522 665 440 665C229 665 44 546 44 318C44 117 193 -15 425 -15C506 -15 593 7 653 21C648 47 647 79 647 115V195C647 262 653 266 703 276V311Z" id="g13-69"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,9.85,0)"><path d="M524 497C519 556 514 633 513 650H40V615C113 608 122 601 122 511V142C122 48 113 42 40 35V0H351V35C264 42 256 48 256 142V300H338C427 300 437 292 448 224H484V423H448C437 359 427 350 338 350H256V571C256 604 260 606 295 606H345C401 606 428 599 449 577C466 559 479 529 489 491L524 497Z" id="g13-68"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="554" vert-adv-y="554"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,17.355,0)"><path d="M559 271C559 395 493 448 433 448C370 448 304 394 275 297C259 245 235 121 219 37C157 54 113 110 113 201C113 310 180 386 276 429L265 457C131 410 23 326 23 186C23 83 86 5 211 -10C180 -159 167 -220 158 -252L165 -261C198 -255 236 -246 251 -218L287 -5C421 9 559 119 559 271ZM484 273C484 144 400 40 293 33L352 331C364 389 382 406 405 406C431 406 484 370 484 273Z" id="g113-253"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="582" vert-adv-y="582"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,28.27,0)"><path d="M574 0L325 548H263L14 0H76L293 479H295L512 0H574Z" id="g117-63"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,39.25,0)"><path d="M703 311H403V276C508 267 513 259 513 189V94C513 63 505 47 490 38C477 29 454 26 434 26C270 26 196 184 196 335C196 515 300 623 420 623C514 623 589 576 613 464L651 472C643 550 638 603 635 637C599 645 522 665 440 665C229 665 44 546 44 318C44 117 193 -15 425 -15C506 -15 593 7 653 21C648 47 647 79 647 115V195C647 262 653 266 703 276V311Z" id="g13-69"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,49.1,0)"><path d="M524 497C519 556 514 633 513 650H40V615C113 608 122 601 122 511V142C122 48 113 42 40 35V0H351V35C264 42 256 48 256 142V300H338C427 300 437 292 448 224H484V423H448C437 359 427 350 338 350H256V571C256 604 260 606 295 606H345C401 606 428 599 449 577C466 559 479 529 489 491L524 497Z" id="g13-68"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="554" vert-adv-y="554"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,56.606,0)"><path d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z" id="g113-245"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="654" vert-adv-y="654"></glyph.data></g></svg></td></tr></table>

<div>Degeneralization of a TGBA corresponding to <svg height="12.6787pt" id="M272" style="vertical-align:-3.571951pt" version="1.1" viewbox="-0.0498162 -9.10675 65.7872 12.6787" width="65.7872pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M703 311H403V276C508 267 513 259 513 189V94C513 63 505 47 490 38C477 29 454 26 434 26C270 26 196 184 196 335C196 515 300 623 420 623C514 623 589 576 613 464L651 472C643 550 638 603 635 637C599 645 522 665 440 665C229 665 44 546 44 318C44 117 193 -15 425 -15C506 -15 593 7 653 21C648 47 647 79 647 115V195C647 262 653 266 703 276V311Z" id="g13-69"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,9.85,0)"><path d="M524 497C519 556 514 633 513 650H40V615C113 608 122 601 122 511V142C122 48 113 42 40 35V0H351V35C264 42 256 48 256 142V300H338C427 300 437 292 448 224H484V423H448C437 359 427 350 338 350H256V571C256 604 260 606 295 606H345C401 606 428 599 449 577C466 559 479 529 489 491L524 497Z" id="g13-68"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="554" vert-adv-y="554"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,17.355,0)"><path d="M559 271C559 395 493 448 433 448C370 448 304 394 275 297C259 245 235 121 219 37C157 54 113 110 113 201C113 310 180 386 276 429L265 457C131 410 23 326 23 186C23 83 86 5 211 -10C180 -159 167 -220 158 -252L165 -261C198 -255 236 -246 251 -218L287 -5C421 9 559 119 559 271ZM484 273C484 144 400 40 293 33L352 331C364 389 382 406 405 406C431 406 484 370 484 273Z" id="g113-253"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="582" vert-adv-y="582"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,28.27,0)"><path d="M574 0L325 548H263L14 0H76L293 479H295L512 0H574Z" id="g117-63"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,39.25,0)"><path d="M703 311H403V276C508 267 513 259 513 189V94C513 63 505 47 490 38C477 29 454 26 434 26C270 26 196 184 196 335C196 515 300 623 420 623C514 623 589 576 613 464L651 472C643 550 638 603 635 637C599 645 522 665 440 665C229 665 44 546 44 318C44 117 193 -15 425 -15C506 -15 593 7 653 21C648 47 647 79 647 115V195C647 262 653 266 703 276V311Z" id="g13-69"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,49.1,0)"><path d="M524 497C519 556 514 633 513 650H40V615C113 608 122 601 122 511V142C122 48 113 42 40 35V0H351V35C264 42 256 48 256 142V300H338C427 300 437 292 448 224H484V423H448C437 359 427 350 338 350H256V571C256 604 260 606 295 606H345C401 606 428 599 449 577C466 559 479 529 489 491L524 497Z" id="g13-68"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="554" vert-adv-y="554"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,56.606,0)"><path d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z" id="g113-245"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="654" vert-adv-y="654"></glyph.data></g></svg> using the standard degeneralization algorithm.</div>

Journal of Applied Mathematics

fig2

Figure 2

Figure 2: Degeneralization Algorithm for Generation of Büchi Automata Based on Contented Situation 